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Cosmological implications of the hydrodynamical phase of group field theory

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Abstract

In this review we focus on the main cosmological implications of the Group Field Theory approach, according to which an effective continuum macroscopic dynamics can be extracted from the underlying formalism for quantum gravity. Within this picture what counts is the collective behaviour of a large number of quanta of geometry. The resulting state is a condensate-like structure made of “pre-geometric” excitations of the Group Field Theory field over a no-space vacuum. Starting from the kinematics and dynamics, we offer an overview of the way in which Group Field Theory condensate cosmology treats solutions for the homogeneous and isotropic universe. These solutions including a bounce, share with other quantum cosmological approaches the resolution of the singularity characterizing general relativity. Contrary to what is usually done in quantum cosmology, in Group Field Theory cosmology no preliminary symmetry reduction is needed for this purpose. We conclude with a discussion of the limits and future perspectives of the Group Field Theory approach.

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Notes

  1. It should be noted that ‘singularity avoidance mechanisms’ may exist in more conventional mini-superspace of quantum geometrodynamics. For instance, from simple particle models like [5] to more comprehensive studies in more realistic situations [6] and recent extension to anisotropic models [7], in which the analysis is consistently based on the behaviour of the wavefunction and not on the bouncing behaviour of quantum-corrected classical equations. For an overview and a comparison between LQC and standard quantum cosmology, see [8].

  2. For further analysis concerning these topics refer to [10].

  3. This procedure is usually generalized to an arbitrary \(k-\)number of massless scalar fields, \(J=0,1,..., k\). Here we restrict the analysis to only four of them because they will be used for labelling the four spatiotemporal dimensions; i.e. 1 temporal \(\phi ^0\) and 3 spatial \(\phi ^i\) independent components.

  4. The fact that \(N\rightarrow \infty \) corresponds to a continuum limit in which a phase transition is reached has been shown in detail in matrix models for two-dimensional gravity [36].

  5. The kinematical Fock space is known to be troublesome for defining an interacting quantum field theory. According to Haag’s theorem one should not expect solutions to the quantum dynamics to be defined as elements on the Fock space.

  6. For a choice that can be generalized [47, 49].

  7. See Sect. 4.2 for clocks and Sect. 5 for rods.

  8. This choice is a convention and can be exchanged; on the other way around, one can start with a right “gauge symmetry” on the \(\varphi \) field and then impose the left invariance over \(\sigma \) obtaining equivalent results.

  9. In the convolution of Wigner \(D-\)matrices with SU(2) intertwiners, the usual range of values for the magnetic indices is taken: \(-j\le m\), \(n\le j\). The indices \(\imath \) labels the possible intertwiners elements in a basis of the Hilbert space; \(\imath _l\) and \(\imath _r\) points to the imposition of the left and right invariance to the field. To a detailed construction of the wavefunction see for instance [38]; here we just sum up the main steps for deriving a cosmological sector from the full theory.

  10. Anyway, in order to translate the theory into a set of equations for cosmological observables, the addition of a scalar field variable is crucial, since it allows us to define within the full theory a set of relational observables with a clear physical meaning. In GFT approaches we find models including a scalar field as matter content. The use of matter reference frames is not new; it dates back at least to DeWitt proposal [103] where coordinates are proposed to be constructed with convenient matter scalar variables (in [104] an extended discussion can be found). More contemporary advances have been obtained by using dust matter to account for this effect. First insights have been proposed by Brown and Kuchař [31] and generalizations to LQG have been developed in Ref. [105]. Relevant advances have been done also by Gielen [60, 97]. With regard to models constructed from the theories discussed in this review, the employment of a massless scalar field as a relational clock defining relational dynamics also appears in canonical LQG [106] and LQC [107, 108].

  11. It would be interesting to contrast the latter deparametrized framework for a single clock with a covariant setting in which one can choose different clocks, following the ideas of [109,110,111]. In this respect, recent work has been done by [112].

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Acknowledgements

We would like to thank Marco De Cesare, Claus Kiefer, Isha Kotecha and Daniele Oriti for fruitful discussions and suggestions. We also thank the referee for their remarks which led to an improvement of the manuscript. This article is part of a project that has received funding from the European Research Council (ERC) under the Horizon 2020 research and innovation programme (Grant agreement No. 758145).

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  1. Institut de Física d’Altes Energies (IFAE), Universitat Autònoma de Barcelona, Campus UAB, Bellaterra, 08193, Barcelona, Spain

    Luciano Gabbanelli

  2. Department of Philosophy, University of Milan, Via Festa del Perdono, 7, 20122, Milan, Italy

    Silvia De Bianchi

  3. Department of Philosophy, Universitat Autònoma de Barcelona, Building B Campus UAB, Bellaterra, 08193, Barcelona, Spain

    Silvia De Bianchi

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Gabbanelli, L., De Bianchi, S. Cosmological implications of the hydrodynamical phase of group field theory. Gen Relativ Gravit 53, 66 (2021). https://doi.org/10.1007/s10714-021-02833-z

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